Finally, we point out that our results for anomalies at disclinations respect the Abelian group structure of topological crystalline phases given by the direct sum
Note that the direct sum operation in the Abelian group structure of topological crystalline phases is applied to both the Hamiltonians \(H_1 \oplus H_2\)  and the representations of its symmetries \(U_1(g) \oplus U_2(g)\) . Here, the Hamiltonians \(H_i\) , \(i = 1,2\) , are elements of the same symmetry class, with corresponding representation \(U_i(g)\)  of all symmetry elements \(g \in G \times G_\text{int}\) .
In particular, in symmetry classes for which the bulk-boundary-defect correspondence holds, the direct sum of a first-order topological phase
with itself cannot lead to a second-order topological phase.
In symmetry classes in which a disclination is connected to a domain wall, meaning that the bulk-boundary-defect classification does not hold, the direct sum of a first-order topological phase with itself may result in a second-order topological phase.
The latter scenario is absent for all cases discussed here.
Our classification results are consistent with corresponding results from Refs. [1]}, [2]}, [3]}.
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